131

BARYON Q-BALLS: A NEW FORM OF MATTER?

STEPHEN B. SELIPSKY Department of Physics Stanford University, Stanford, California 94305

Abstract Hadronic effective field theories describing ordinary nuclei also contain ·Q-Ba.11 solu­ tions which can describe a new state of matter, "baryon matter" . Baryon matter is stable in very small chunks as well as in stellar-sized objects, since it is held together by the strong force instead of just . Larger chunks, "Q-" , in which gravity is impor­ A tant, model neutron stars. wide variety of Q- models, a.II consistent with known nuclear physics, allow compact objects to have masses much larger, or rotation periods much shorter, than is conventionally believed possible. Smaller chunks of nuclear density baryon matter could also be astrophysically important components of the universe, a.nd at late times would have many properties similar to those of strange matter chunks. 132

1. Introduction

There are many surprising possibilities lurking in the non- perturbative sector of field theories. Here we add to the body of work on this topic, presenting the possibility of a new state of matter which a.rises from the discovery of solutions to effective field theo­ ries describing among other things ordinary nuclei. Effective field theories of interacting baryons and mesons ha.ve successfully reproduced measured properties of nuclei a.s wella.s 1 6 results of scattering experiments -3). We have found4- ) non-topological classical solu­ tions to such theories: fermion Q-Balls, or Q-Sta.rs in the ca.se where their self-gravity is important and gravitational effects are included. Q here stands for the conserved charge

(baryon number) which stabilizes the matter against decay. The properties of such a state of baryon matter can be different from and essentially independent of the characteristics of ordinary nuclei studied in the laboratory. In particular, neutron stars may have a large binding energy per nucleon, hundreds of MeV, due mainly to nuclear forces, and may be 7 8 more massive or able to rotate fa.st.er than is suggested by currently accepted limits. ' )

1 In addition, chunks of baryon matter, varying in size from 10- 2cm to several kilometers, may also exist. Structural characteristics are fairly insensitive to the effective field theory used; classes of theories give the same equation of state.

2. Constructing Solutions

There is a simple graphical method for finding large baryon-number solutions. Con­ sider a. Lagrangian for a baryon 1jJ intera.cting with a. sca.la.r a.nd vector field:

(2.1)

1 It is a good a.pproxima.tion to neglect the dynamics of the vector field ), and fora many­ fermion system we can make the Thomas-Fermi approximation (dm/dr � m2) to get a Fe rmi sea of baryons described by a constant chemical potential EF and a Fermi momen­ 6 tum kF slowly varying in space. The identity gives the cla.ssica.l (i}ij;) = -(8Pv, /8m) ) 133

equation of motion

(2.2)

where P.p is the pressure of the baryons. This is equivalent to Newton's ma for a. F = · mechanical 'particle' a.t 'position' a a.t 'time' r moving in a potential Veff = P.p - U, with

'friction' from the (2/r)&a/&r term negligible for large r.

The Q-Ball solution occurs when a rolls between degenrra.t.r maxima of V.Jf, where

one of the maxima is the vacuum and the height. of I.he other is tuned by varying The Ep. a fieldstarts at a value a inside at r = 0, close to the top of t he first hill, and stays near I.hat value out to a. large radius (since the I.op of the hill is flat) before quickly rolling to the

vacuum. The large baryon number Q-Ba.ll thus has a flat interior, a radius which is a free

para.meter, and a thin surface (of ordrr the scalar Compton wavelength). The density

is determined by algebraic equations for and ainsideo requiring degenerate maxima.: Ep Ve ff = 0 and &V,JJ/&a = 0. Q-Ba.11 must still have total energy E QmN in order to A < be bound. Further details can be found in Refs. 6 and 9.

3. Application

We can apply this construction quite generally, for instance to Wa.lecka's Quantum 1 Hadro-Dynamics ), in which the proton and neutron a.re fermion fields 1/'i whose effective

mass is m(a) = g,a, with the free-particle g8a0. The scalar field has a quadratic mN = potential U = m;(a - a0 )2, and the vector field is the w particle. This theory's large ! baryon number Q-Ball corresponds to infinite symmetric nuclear matter when N = Z. The small baryon number solutions (nuclei) depend on the friction term in eq. (2.2); the

rolling starts above P.p - U = 0 and friction brings a to rest at the vacuum value where

m = mN. Fig. 1 translates the Walecka. solutions into this framework.

Infinite nuclear matter is desta.bilizrd by Coulomb forces. In fact, neutrons, protons,

and electrons coexist in ,8-decay equilibrium, allowing local charge neutrality: kp,1, = kF,e 134

->e.... •ca ...... , / ' . · 0.2 , ' /·�- ··.�l ' �� . . . ' u. ·. · b . � ' ... . " 01 · ."::::�' \ I \ ' I ' ' I � ' 2 I r., \ Q�� 0.0

-0.1 a.• o.• o.• -0.2'1 0.25 0.5 0.715 m/m• m/m.

1. Figure Graphical representation of Q-Balls in a nuclear theory. U(u) (dots) is the scalar field potential, is the baryon pressure, and U is the effective potential that 'rolls' in, P.p P.µ - u starting at r = 0 (marked by crosses). Figure 2. Potentials U (solid lines) given in Refs. 1 and 2.

and E:F n E:F E:F e With a baryon pressure function reflecting this, a large baryon , ,p , · = + number Q-Ba.11 does not exist in Wa.lecka.'s theory, so strong gravity for stellar sized objects is the only possibility. However we a.re still free to choose other potentials. In an effective field theorythe potential and the coupling m(a) a.re in genera.I nonrenormaliza.ble, subject only to the symmetry of the underlying theory, and should either be fit fromexperiment (a la Walecka) or derived from the underlying theory (QCD). Then m and U have quantum corrections included in their definitions. Bogut.a., Strocker, and others have found many models with renorma.liza.hle potentia.ls which reproduce known hulk properties of nuclear matter\ Fig. 2 shows these and Wa.lecka.'s potentials; the region 'a.-b-c-d'is the only pa.rt relevant for nuclear data, and U is not constrained elsewhere, allowing infinitely many altered potentials (e.g. dotted lines) that do give la.rge baryon number Q-Balls. Only the

(currently unmeasurable) point determines Q-Ball hulk properties, and the rest of U ' f ' affects only the thin surface. Iu this way we can have stable macroscopic chunks of high density baryon matter, of any size above where surface dfPcts become important. Qbulk

To fix point 'f' and the properties of nn1tron stars, experiments on baryon matter chunks would be necessary. 135

The simple 'chiral' Q-Ball case in which the bulk-phase nucleon mass m(ainside) = 0,

has features genetic to more complicated models (which have also been solved). The

equation of state9) for a chiral Q-Star (neglecting the electron mass and neutron-proton mass difference) depends only on the vector repulsion strength and Uo = U (a inside):

3/0 E-3P-4Uo + o:v (E-P-2Uo) -=0, (3.1)

2 112 7r to the strong forces, so charge with /mv ) 3 / . Only the nucleons couple O:v = (gv separation results in the surface. Structure calculations may neglect these details since

the surface width is about 10-20 times sma.ller than the stellar radius, but the electrostatic

gap means that small Q-Balls will not absorb ordinary matter and Q-Stars can support

a crust of degenerate ordinary matter below neutron drip densities.

Since Q-Balls are a bulk phase with an equation of state\ gravity can be included

simply by integrating the Oppenheimer-Volkoffequations. On dimensional grounds, when

GM/R 1 a Q-Star has radius R � mvw0'2 � 100 km and mass M � m;1a02 10 M0, � � where IOOMeV is the vacuum val ue of Integrating the Oppenheimer-Volkolf a0 � a. equations for eq. (3.1) gives the solid lines in Fig. 3. We also show for comparison

(dashed lines) two neutron sta.r models (pion condensation and Walecka's) which are

respectively among the least stiff a.nd most stiff of conventional models. In

conventional models the stellar radius decreases as the mass and baryon number increase,

while Q-Balls have M and Q giving the generic curlique. ex R3,

Both stellar radius a.nd mass increase as the unknown Uo decreases, allowing a large

upper limit for the neutron star mass if Uo is small. However eq. (3.1) is only valid

at densities high enough for the nuclear interactions to domina.te. There seem to be no experimental constraints on the equation of st.ate of a large nvmber of baryons above

densities 0 0gm cm3 which we will take as the lowest permissible ( I / ), U0 < 1 val ue. Using (3.1) down to density Uo evades the theoretical Rhoades-Ruffini limit, 3.2M8,

which assumes some conventiona.l <"qna.tion of st.ate below a density ta.ken to be about [0 136

x trI ]

10 20 30 •o R(km) 3. Mass stellar radius for chiral (solid line�) au

For 1.23 (Walecka 's value ll"hen the upper mass limit U0 4 a.U�/2 = (100MeV) ), on a neutron star with equation of state (:l.J) is

6' �-1 , (:3.2) fr'1 ma r: ,.-l/2 D''1 x = l). ::..; -'100 where U100 = U0/(lOOJ\1eV)4. The star a.lso reaches its maximum baryon number at this Uo, mass: Qmax = 22.5 U�3j4 57 Th" scaling with including R holds only x 10 . � u-;;1 if is simultanpously sea.led as for bound configurations without this!2, sea.ling, Civ u-;;1!2; km 0.48M0 < Mmax < •100M,o.; 2.8 < Rmax < 2300 km 2 2 0.67 < < 1.7 g cm < < 2. g cm 1057 Qmax 1060; :3 .-1 1043 Imax 0 1052 x x x x (3.3)

Even small Q-Balls are stable against dispersal into free particles, because of their strong-interaction binding energy, and against. adiabatic radial pulsations, since their 1 adiabatic index U0(Pt1, + P, U)- 1/:3. The turning point for the stability of r � - » this mode occurs at the point when' 0, indica.t.Pd by crosses in Fig. 3. di\!/dE central = so Q-Stars before the maximum mass in Fig. 3 ane stable. For a rotating Q-Star, the 8 relativistic Maclaurin spheroid analysis as applied ) to other neutron star models shows 137

" that dense Q-Stars can rotate at extrcnwly high rates ), easily above the 0.5 ms limits on other equations of st.ate. At the high densities needed, an effective field theory of baryons and mesons might not be applicable, but quark Q-Star models might apply. The chiral Q-Ball equation of state (negl<>ct.ing wet.or repulsion) has the form ofthe MIT bag 10 model's strange stars and nugg<>ts ) (neglecting gluon exchange). The potential U(a) dynamically generates the analogtw of bag pressure, but the of Uo is of course unrelated to the nucleon bag pressure.

The Q-Sta.r model differs in important. respects from conventional neutron sta.r models, raising interesting astrophysical qtwstions. In addition to the stella.r structure limits discussed above, electromagnetic properties of Q-Stars might be very different from those of neutron stars in conventional models. The neutrino cooling rate should also be much higher, since there is a boson condensate to absorb momentum for the first-order URCA process. Only a restricted class of <>ff<'d i\"e field theories in which U wiggles such that there are two separate Q-Star pha-'es (with different chemical potentials) can describe both neutron stars with large masses (� :3,U,,,) and neutron stars with short rotation periods 0.5ms). A conventional neutron star phase is always also present in any field (Prot < theory which contains Q-Stars, and which phase is preferred will depend on the theory. 11 In the early universe, a hadronic phase transition may well produce Q-Ba.lls ), which

(unlike strange nuggets) can be deeply enough bound to survive evaporation for 1035. Q 2: #1 Q-Balls absorb neutrons and inhibit. nucleosynt.hesis. In the present universe, baryonic dark matter can reside in remaining Q-Balls; many st.range matt.ercalcul a.tions still apply 12 to the observability of a cosmic or galactic flux ), since tlwse a.re also approximately nuclear-density lumps with Coulomb barrier to fusion. For the larger Q-Ba.lls that a. survive evaporation, direct detection would be difficult. However Q-Ba.lls have quite a low surface tension and tend to break up into smaH droplets in any violent event. Neutron star collisions or possibly explosions could produce some ga.la.ct.ic flux of smaller

thank Jes Madsen fordiscussions on point.s. #1 I t.lwse 138

Q-Balls.

Acknowledgements: I thank tlw organizers for an enjoyable and stimulating conference, and the CERN Theory Group for its hospitality from January through June this year.

This work was done in collaboration with Bryan Lynn and Safi Bahcall, and was supported in part under a U.S. National Science Fo undation Graduate Fellowship.

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